p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.278C23, C22⋊C4○SD16, (C4×C8)⋊47C22, C4⋊C4.401D4, C4.102(C4×D4), (C4×Q8)⋊2C22, (C2×SD16)⋊10C4, SD16⋊11(C2×C4), (C4×SD16)⋊51C2, C8.21(C22×C4), C4.26(C23×C4), D4.9(C22×C4), C22.69(C4×D4), C8⋊C4⋊39C22, C2.D8⋊67C22, C4.Q8⋊74C22, SD16⋊C4⋊3C2, Q8.8(C22×C4), C4⋊C4.366C23, C8○2M4(2)⋊7C2, (C2×C8).417C23, (C2×C4).206C24, C22⋊C4.188D4, C2.6(D4○SD16), (C4×D4).58C22, C23.438(C2×D4), Q8⋊C4⋊93C22, (C2×D4).375C23, (C2×Q8).347C23, (C22×SD16).4C2, C23.25D4⋊25C2, C23.38D4⋊32C2, C22.11C24.7C2, (C22×C8).249C22, (C22×C4).927C23, C22.150(C22×D4), D4⋊C4.197C22, C23.32C23⋊6C2, C42⋊C2.83C22, (C2×SD16).111C22, C23.37D4.10C2, (C22×D4).323C22, (C22×Q8).259C22, (C2×M4(2)).353C22, C2.66(C2×C4×D4), (C2×C8)⋊15(C2×C4), (C2×Q8)⋊21(C2×C4), C4.14(C2×C4○D4), (C2×C4).913(C2×D4), (C2×D4).138(C2×C4), (C2×C4).265(C4○D4), (C2×C4).265(C22×C4), SmallGroup(128,1681)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.278C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=b2c, de=ed >
Subgroups: 436 in 246 conjugacy classes, 140 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22×C8, C2×M4(2), C2×SD16, C22×D4, C22×Q8, C8○2M4(2), C23.37D4, C23.38D4, C23.25D4, C4×SD16, SD16⋊C4, C22.11C24, C23.32C23, C22×SD16, C42.278C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, D4○SD16, C42.278C23
►On 32 points
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 19 15 5)(2 20 16 6)(3 17 13 7)(4 18 14 8)(9 22 25 30)(10 23 26 31)(11 24 27 32)(12 21 28 29)
(1 4 3 2)(5 18 7 20)(6 19 8 17)(9 29 11 31)(10 30 12 32)(13 16 15 14)(21 27 23 25)(22 28 24 26)
(1 31 15 23)(2 32 16 24)(3 29 13 21)(4 30 14 22)(5 10 19 26)(6 11 20 27)(7 12 17 28)(8 9 18 25)
(1 3)(2 14)(4 16)(5 7)(6 18)(8 20)(9 27)(10 12)(11 25)(13 15)(17 19)(21 23)(22 32)(24 30)(26 28)(29 31)
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,19,15,5)(2,20,16,6)(3,17,13,7)(4,18,14,8)(9,22,25,30)(10,23,26,31)(11,24,27,32)(12,21,28,29), (1,4,3,2)(5,18,7,20)(6,19,8,17)(9,29,11,31)(10,30,12,32)(13,16,15,14)(21,27,23,25)(22,28,24,26), (1,31,15,23)(2,32,16,24)(3,29,13,21)(4,30,14,22)(5,10,19,26)(6,11,20,27)(7,12,17,28)(8,9,18,25), (1,3)(2,14)(4,16)(5,7)(6,18)(8,20)(9,27)(10,12)(11,25)(13,15)(17,19)(21,23)(22,32)(24,30)(26,28)(29,31)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,19,15,5)(2,20,16,6)(3,17,13,7)(4,18,14,8)(9,22,25,30)(10,23,26,31)(11,24,27,32)(12,21,28,29), (1,4,3,2)(5,18,7,20)(6,19,8,17)(9,29,11,31)(10,30,12,32)(13,16,15,14)(21,27,23,25)(22,28,24,26), (1,31,15,23)(2,32,16,24)(3,29,13,21)(4,30,14,22)(5,10,19,26)(6,11,20,27)(7,12,17,28)(8,9,18,25), (1,3)(2,14)(4,16)(5,7)(6,18)(8,20)(9,27)(10,12)(11,25)(13,15)(17,19)(21,23)(22,32)(24,30)(26,28)(29,31) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,19,15,5),(2,20,16,6),(3,17,13,7),(4,18,14,8),(9,22,25,30),(10,23,26,31),(11,24,27,32),(12,21,28,29)], [(1,4,3,2),(5,18,7,20),(6,19,8,17),(9,29,11,31),(10,30,12,32),(13,16,15,14),(21,27,23,25),(22,28,24,26)], [(1,31,15,23),(2,32,16,24),(3,29,13,21),(4,30,14,22),(5,10,19,26),(6,11,20,27),(7,12,17,28),(8,9,18,25)], [(1,3),(2,14),(4,16),(5,7),(6,18),(8,20),(9,27),(10,12),(11,25),(13,15),(17,19),(21,23),(22,32),(24,30),(26,28),(29,31)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4X | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | D4○SD16 |
| kernel | C42.278C23 | C8○2M4(2) | C23.37D4 | C23.38D4 | C23.25D4 | C4×SD16 | SD16⋊C4 | C22.11C24 | C23.32C23 | C22×SD16 | C2×SD16 | C22⋊C4 | C4⋊C4 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C42.278C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 1 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 16 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 0 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 | 5 |
| 0 | 0 | 12 | 5 | 5 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[13,2,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,16,0,0],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,10,5,0,12,0,0,7,7,5,5,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
C42.278C23 in GAP, Magma, Sage, TeX
C_4^2._{278}C_2^3 % in TeX
G:=Group("C4^2.278C2^3"); // GroupNames label
G:=SmallGroup(128,1681);
// by ID
G=gap.SmallGroup(128,1681);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2019,521,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=b^2*c,d*e=e*d>;
// generators/relations